Let f : X → Y be a closed
proper surjection whose Leray sheaves are locally constant through a given dimension
k. Spectral sequences are used to analyze the cohomological connectivity and
manifold properties of Y . Generally, when Y has dimension at most k, it is a
cohomology k -manifold over a given principal ideal domain R if and only if
Hq(X,X −f−1y;R) is isomorphic to HomR(Hq−k(f−1y;R),R) for every y ∈ Y and
q ≤ k. As a result, if X is an orientable (n + k)-manifold, each f−1y has the shape of
a closed, connected, orientable n- manifold, and Y is finite dimensional, then Y is a
generalized k-manifold. Euler characteristic relationships involving X, Y ,
and the typical fiber f−1y are derived in case the Leray sheaves of f are
locally constant in all dimensions and the range has cohomologically finite
type.
